Incomplete generalized U Statistics for food risk assessment

De
PATRICE BERTAIL

25
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Incomplete generalized U-Statistics for food risk assessment. Patrice Bertail CREST, Laboratoire de Statistique Jessica Tressou INRA, Laboratoire de recherche sur la consommation Abstract : This paper proposes statistical tools for quantitative evaluation of the risk due to the presence of some particular contaminants in food. We focus on the estimation of the probability of the exposure to exceed the so-called provisional tolerable weekly intake (PTWI), when both consumption data and contamination data are independently available. A Monte-Carlo approximation of the plug-in estimator, which may be seen as an incomplete generalized U-statistics, is investigated. We obtain the asymptotic properties of this estimator and propose several con?dence intervals, based on two estimators of the asymptotic variance: (i) a bootstrap type estimator (ii) an approximate jackknife estimator relying on the Hoe?ding decomposition of the original U-statistics. As an illustration, we present an evaluation of the exposure to Ochratoxin A in France. Résumé : Cet article propose des outils statistiques d?évaluation du risque d?exposition due à la présence de certains contaminants dans l?alimentation. Nous cherchons essentiellement à estimer la probabilité que l?exposition dépasse la dose toxicologique hebdomadaire tolérable, lorsqu?on dispose de données de consomma- tion et de données de contamination indépendantes. On propose une approxima- tion de type Monte-Carlo de l?estimateur empirique de cette quantité, s?écrivant comme une U-statistique généralisée incomplète.

1 Introduction Food may be naturally contaminated by some chemical components which may become toxic for the human organism if the total amount ingested through food consumption exceeds a certain tolerable dose. For example, Ochratoxin A (OTA) is a natural mycotoxin produced by fungi of the As-pergillus and Penicillium families , which has been classied as a genotoxic carcinogen in 1998 by the European Scientic Committee for Food. It may be detected in many products including cereals, grapefruit, dry fruits or veg-etables, wine, co¤ee, beer, or pork and poultry meat. An important toxicological concept to measure the medical impact of a contaminant is the so called Provisional Tolerable Weekly Intake (PTWI) expressed in terms of nanogram per body weight per week (ng/kgbw/wk in the following). It is xed in Europe at 35 ng/kgbw/wk for OTA. This quantity is the scientically and medically recognized level over which a permanent excess may be considered as potentially dangerous for the hu-man health (without any distinction between individuals except their body weight). Even though its value may not be the same for di¤erent countries, this quantity generally serves as the basis to decide whether or not there is a specic public health problem related to a particular contaminant and to plan food regulatory programs. In particular, an important issue is to evaluate whether the (complete or partial) suppression of the contaminated products or the reduction of the contamination in some product (for instance by imposing a maximal limit to certain commercialized items) may have a signicant impact on the global exposure of the individuals. Our approach in this study will be to evaluate the probability that the individual exposure over a week exceeds the PTWI. This view is not com-pletely satisfactory from a medical point of view, because it does not take into account for the dynamic of the contamination and exposure phenom-enon. Actually because of the lack of data, the permanent exposure over a lifetime is di¢ cult to estimate, thus our parameter may rather be interpreted as the probability of occasional short-term excursions above the PTWI than a true probability to develop a disease because of the exposure to the con-taminant. However, it still remains an important indicator: this is actually the main risk indicator which is currently used in international committee (see Codex Alimentarius, website). Estimating precisely its value and giving condence intervals is thus of prime importance. From a statistical point of view, if one could observe in a survey the global individual exposure dened as the quantity of contaminant ingested on a certain period relative to the body weight of the individual, one could estimate the mean of global exposure or the probability of the exposure (over a given period of observation) to exceed the PTWI. Such data are currently not available since it would involve repeated costly chemical analysis of all